Mechanical impedance of a structure or material, in general, is defined as the quotient of the excitation force or moment, and the caused velocity response along the same axis of excitation. If the excitation and motion are at the same point, it is a point impedance. If not, it is transfer impedance. If displacement or acceleration is used instead of velocity, the quotient is the dynamic stiffness, or inertiance, respectively. These quotients are often termed as Frequency Response Functions (“FRF”s) as they are complex functions expressed in spectra of modulus and phase. FRF's describe the inherent dynamic characteristics of the test structure or material and do not change with external excitation. They therefore are commonly utilized to characterize the dynamic behavior of a structure or material.
The motion at a point of an object in space is fully described by six degrees of freedom: three translational and three rotational. Point translational impedances are the ratio of the force applied to the structure to the linear velocity along the force direction, measured at the same point. Point rotational impedances are the ratio of the moment applied to the structure to the angular velocity along the moment direction, measured at the same point. These ratios are normally acquired in modal analysis and material characterization, and are widely utilized to facilitate modern structural and mechanical design, sound and vibration control, and so forth.
When measuring the translational impedance at a single point along a direction, the most commonly used technique is to excite the test structure using a shaker; and to measure the excitation force and the output motion by a conventional load cell and an accelerometer, respectively. This means that three devices are involved: a shaker, a force sensor, and a motion sensor. Amplifiers are often required for each of these devices. The loading effects, calibration errors, and misalignments introduced by the three devices alone or together often affect the accuracy of measurement. In addition, if on-site installation of exciters and sensors on structures is necessary, ensuring true measurement at the driving point and along the same axis could be a major error-source that would need to be eliminated through time-consuming procedures. In cases where the object under testing is small in size, these problems are amplified, and more difficult to overcome.
Compared to translational impedance, measuring rotational impedance is more difficult, and more inaccurate. Due to the lack of proper transducers for creating moments, and measuring rotational motion, only translational impedances are able to be identified in conventional dynamic testing. After tests are completed, all rotational motions are interpolated from the translational motions at two or more adjacent points. In numerical simulations, both translational and rotational impedances are able to be utilized to quantify the characteristics of a dynamic system.
Many studies have attempted to improve the technology of detecting rotational impedance, or other rotational FRFs. To test for rotational impedance, three transducers were required: a pure moment exciter, a moment sensor to detect the applied moment, and a motion sensor to measure the resultant angular motion. All three transducers do not exist and the tests are normally done indirectly using linear force and motion transducers.
FIG. 1 shows a prior art apparatus. A pair of identical shakers 10 is operated in opposing phases to provide moment excitation to the test structure through a lever 12. Two identical load cells 14 are placed on the lever 12 to measure the excitation forces from the shakers 10 and consequently to deduce the moment exerted to the structure 11. The resultant rotational motions are then measured by a pair of identical accelerometers attached onto the structure surface close the receiver point 15 of the test structure. This two-shaker approach possesses severe problems leading to large measurement errors. The principal problem is the dependence of the measurement on the cross mobility at the receiver points 15. In a structure, most points, except the symmetric center, have non-zero cross mobility which simply means a force or moment excitation along a direction also generates translational and rotational motions in other directions. When the connection point of the whole apparatus to the structure is not the symmetric centre, the cross mobility of the structure would cause errors as the exerted moment may generate linear forces or a moment along other axes which do not contribute to rotational velocity under measurement. These errors due to cross mobility are too severe to be neglected as the impedances are defined as the quotient of excitation and caused response along the same axis. Eliminating the errors is by use of additional complex correction items which can only be quantified through further FRF measurements.
The two-shaker approach is further troubled by its very limited frequency range of excitation. This limit is caused by the vibration of the lever 12 connecting the shakers 10 to the test structure 11. To validate the basic assumption of the approach that the lever 12 is rigid, the method can only be employed in a frequency range far below the first natural frequency of the lever 12.
The third problem is not less significant: the pair of load cells 14 and the pair of linear accelerometers 13 must be very similar, if not identical. Tiny differences in specifications or performance could cause major errors as the measurements are based on the differences between the outputs of the two sensors. Lastly, the size and complexity of the apparatus make its use and installation inconvenient and difficult.